What does it mean for AlphaZero's network to be "fully trained"
Neural network will eventually reach limit of it's approximation power. You can't expect to learn more and more things infinitely long with the same amount of learnable parameters. Also, if you eventually reach optimal performance, you can't play more optimal than what optimal is (not saying that it reached optimal performance but possibly something close to optimal for it approximation abilities). So probably combination of those two leads for the performance increase to reach it's limit.